Linear Algebra

Section II. Similarity 361⋆ ⋆ ⋆⋆⋆⋆⋆ ⋆ ⋆. . .To develop a canonical form for representatives of the similarity classes wenaturally build on previous work. This means first that the partial identitymatrices should represent the similarity classes into which they fall. Beyondthat, the representatives should be as simple as possible and the partial identitiesare simple in that they consist mostly of zero entries. The simplest extension ofthe partial identity form is the diagonal form.2.1 Definition A transformation is diagonalizable if it has a diagonal representationwith respect to the same basis for the codomain as for the domain. Adiagonalizable matrix is one that is similar to a diagonal matrix: T is diagonalizableif there is a nonsingular P such that PTP −1 is diagonal.2.2 Example The matrix ( )4 −21 1is diagonalizable.( ) ( ) ( ) (2 0 −1 2 4 −2 −1 2=0 3 1 −1 1 1 1 −12.3 Example This matrix is not diagonalizable( )0 0N =1 0) −1because it is not the zero matrix but its square is the zero matrix. The fact thatN is not the zero matrix means that it cannot be similar to the zero matrix,by Example 1.3. So if N is similar to a diagonal matrix D then D has at leastone nonzero entry on its diagonal. The fact that N’s square is the zero matrixmeans that for any map n that N represents, the composition n ◦ n is the zeromap. The only matrix representing the zero map is the zero matrix and thus D 2would have to be the zero matrix. But D 2 cannot be the zero matrix becausethe square of a diagonal matrix is the diagonal matrix whose entries are thesquares of the entries from the starting matrix, and D is not the zero matrix.That example shows that a diagonal form will not suffice as a canonicalform — we cannot find a diagonal matrix in each matrix similarity class. However,the canonical form that we are developing has the property that if a matrix canbe diagonalized then the diagonal matrix is the canonical representative of itssimilarity class.